For the solution of large sparse nonnegative constrained least squares (NNLS) problems, a new iterative method is proposed which uses the CGLS method for the inner iterations and the modulus iterative method for the outer iterations to solve the linear complementarity problem resulting from the Karush-Kuhn-Tucker condition of the NNLS problem. Theoretical convergence analysis including the optimal choice of the parameter matrix is presented for the proposed method. In addition, the method can be further enhanced by incorporating the active set strategy, which contains two stages where the first stage consists of modulus iterations to identify the active set, while the second stage solves the reduced unconstrained least squares problems only on the inactive variables, and projects the solution into the nonnegative region. Numerical experiments show the efficiency of the proposed methods compared to projection gradient-type methods with less iteration steps and CPU time. We also apply the proposed method to image restoration with Tikhonov regularization, and show that the proposed method gives clearer images compared to previous methods.

This is joint work with Dr. Ning Zheng and Prof. Jun-Feng Yin